Semialgebraic Convex Hulls via SDP Projection

Updated 29 March 2026

Semialgebraic convex hulls are the closed convex envelopes of semialgebraic sets, represented as projections of spectrahedra defined by linear matrix inequalities.
The SDP projection method leverages moment-SOS hierarchies and duality techniques to yield explicit, optimal semidefinite representations with controlled extension degrees.
This approach underpins applications in polynomial and quadratic optimization, offering computational benefits and deeper insights into convex hull complexity.

A semialgebraic convex hull via SDP projection is a convex set obtained as the closure of the convex hull of a semialgebraic set, described or approximated by projecting a spectrahedron (a solution set to an LMI) through a linear map. This area intersects real algebraic geometry, convex analysis, and semidefinite optimization. The study of when and how such convex hulls admit explicit semidefinite representations, what complexity arises in their LMI description (measured via the semidefinite extension degree), and exactness phenomena for various algebraic and semialgebraic sets are central themes.

1. Semialgebraic Convex Hulls and SDP Projections

Given a semialgebraic set $S \subseteq \mathbb{R}^n$ , its closed convex hull $K = \overline{\operatorname{conv}}(S)$ often arises in polynomial and quadratic optimization. A set $K \subseteq \mathbb{R}^n$ is a spectrahedral shadow if there exist real symmetric matrices $M_0, M_1, \dots, M_n, N_1, \dots, N_m$ of size at most $d$ and a linear map $\pi : \mathbb{R}^{n+m} \to \mathbb{R}^n$ such that

$K = \pi\left( \left\{ (x, y) \in \mathbb{R}^n \times \mathbb{R}^m : M_0 + \sum_{i=1}^n x_i M_i + \sum_{j=1}^m y_j N_j \succeq 0 \right\}\right).$

The minimal matrix size $d$ for such a representation is the semidefinite extension degree $\mathrm{sxdeg}(K)$ , an intrinsic invariant quantifying the complexity of semidefinite optimization over $K$ (Scheiderer, 2024). Spectrahedral shadows include polyhedra ( $\mathrm{sxdeg}=1$ ), second-order cone representable sets ( $\mathrm{sxdeg}=2$ ), and more general convex semialgebraic sets.

2. Semidefinite Extension Degree and Optimal Bounds

The semidefinite extension degree measures the minimal block-size in LMI representations of $K$ . It is known that:

$\mathrm{sxdeg}(K) = 1$ iff $K$ is a polyhedron.
$\mathrm{sxdeg}(K) = 2$ iff $K$ is second-order-cone representable.
For arbitrary closed convex semialgebraic $K \subset \mathbb{R}^2$ , $\mathrm{sxdeg}(K) \le 2$ (Scheiderer, 2020).
For the closed convex hull $K$ of any $\dim 1$ semialgebraic set $S \subset \mathbb{R}^n$ , $\mathrm{sxdeg}(K) \le 1+\lfloor n/2\rfloor$ (Scheiderer, 2024, Scheiderer, 2012). This bound is sharp for monomial and rational normal curves.

The extension degree governs the size of the PSD matrices needed in practical optimization and is thus a critical computational parameter.

3. Explicit SDP Lifts: Curves, Varieties, and Rational Parameterizations

Extensive theory describes when explicit SDP lifts are achievable:

For any $1$-dimensional semialgebraic $S \subseteq \mathbb{R}^n$ , the moment-SOS hierarchy or Schur–Hankel determinant techniques yield an exact SDP projection of size $1+\lfloor n/2\rfloor$ (Scheiderer, 2024, Scheiderer, 2012, 0901.1821).
For rational varieties (especially curves and low-degree hypersurfaces), the convex hull is given by the projection of a moment matrix cone $M_d(y) \succeq 0$ $M_{d} (y) ⪰ 0$ , with affine constraints modeling the parameterization (0901.1821). This applies for:
- Curves ( $m=1$ parameters).
- Quadratically-parameterized hypersurfaces ( $d=1$ ).
- Bivariate quartic hypersurfaces ( $m=d=2$ ).

For each, an explicit (block-Hankel) LMI representation is available, and linear optimization over the convex hull reduces to an SDP in the lifted variables.

4. Proof Techniques: Duality, Sums-of-Squares, and Moment Relaxations

The construction of SDP lifts typically leverages:

Moment-SOS hierarchies: For a variety $V \subseteq \mathbb{R}^n$ defined by $g_i(x)=0$ , moment matrix $M_d(y)\succeq 0$ and (for constraints) localizing matrices $M_{d-d_i}(g_i\,y)\succeq 0$ are imposed over truncated moment sequences $y_\alpha$ (Rostalski et al., 2010, Gouveia et al., 2010). The convex hull is identified as the projection onto $(x_1,\dots,x_n)$ of the spectrahedron defined in the $y$ -space.
Algebraic characterizations: The semidefinite extension degree is characterized via sum-of-squares decompositions of rank-constrained tensor evaluations. In the curve case, Schur/Hankel determinant expansions yield the required bounds on the rank (Scheiderer, 2024).
Facial structure and dual cones: Facet-exposure methods and dualization link LMI lifts to the cone of nonnegative functionals on the underlying set.

For QCQPs, analysis of the convex hull of the epigraph uses the geometry of the cone $\Gamma$ of convex Lagrange multipliers, leading to explicit LMI formulations when exactness holds (Wang et al., 2024, 2002.01566).

5. Special Cases, Applications, and Algorithmic Implications

Several cases admit more efficient or alternative representations:

In $\mathbb{R}^2$ , every closed convex semialgebraic set has $\mathrm{sxdeg} \leq 2$ and can be represented using second-order cones; the tangent-tensor approach provides explicit SOCP lifts (Scheiderer, 2020).
Convex hulls of curves and points: For finite unions and certain higher-degree cases finite convergence occurs in the theta-body (theta hierarchy) or truncated moment hierarchy (Gouveia et al., 2010).
The spectrahull $SH(S) = \{ (\operatorname{Tr}(A_1 X), ..., \operatorname{Tr}(A_m X)) : X \in \Delta_n \}$ (spectraplex) generalizes classical convex hulls, with membership problems reducible to SDP feasibility and addressed via iterative algorithms (e.g., the Triangle Algorithm) (Kalantari, 2019).
In quadratic programming, the convex hull of QCQP epigraphs coincides with the standard SDP relaxation under verifiable geometric symmetry, such as high quadratic eigenvalue multiplicity or simultaneous diagonalizability of data matrices (Wang et al., 2024, 2002.01566).

The effective block-size has direct algorithmic implications for interior-point solvers, with smaller $\mathrm{sxdeg}$ enabling more efficient optimization.

6. Limitations, Obstructions, and Open Problems

There are critical limitations:

For higher-dimensional ( $\dim \ge 2$ ) semialgebraic sets, there exist natural convex sets not representable as spectrahedral shadows (Scheiderer, 2024). The one-dimensional case represents a boundary for semidefinite representability.
The theta-body relaxation does not always achieve finite convergence except for finite point sets or when no so-called "convex-singular" points appear (Gouveia et al., 2010).
In general, for varieties of high dimension or degree, the convex hull may not be obtainable by a semidefinite projection of feasible size due to sum-of-squares limitations (e.g., Motzkin's polynomial not being a sum of squares).
In the noncommutative (free) setting, matrix convex hulls of free semialgebraic sets are approximated by hierarchical outer SDP relaxations, but are rarely themselves spectrahedral shadows or "semialgebraic" in the free sense (Helton et al., 2013).
The question of whether every convex semialgebraic set is a spectrahedral shadow (Helton–Nie Conjecture) is open in full generality, though it is equivalent to the statement that convex hulls of rank-one elements of any spectrahedron are spectrahedral shadows (Harrison, 2015).

The functional/duality approach and moment-SOS techniques are essential threads:

Moment/SOS hierarchies yield systematic outer approximations to $\operatorname{conv}(V)$ —the intersection of spectrahedral shadows converges (sometimes finitely, sometimes asymptotically) to the convex hull (Rostalski et al., 2010, Gouveia et al., 2010).
Duality links (KKT, convex geometric, and projective duality) connect SDP relaxations to underlying algebraic geometry and the structure of the boundary of convex hulls (Rostalski et al., 2010).
Explicit Hankel matrix representations are central for rational normal and monomial curves, providing sharp extension degree bounds (Scheiderer, 2024).

A paradigmatic instance is the convex hull of a rational normal curve in $\mathbb{R}^{n+1}$ . Let $RNC_n = \{ (t, t^2, \ldots, t^n) : t \in \mathbb{R} \}$ . The closed convex hull $\overline{\operatorname{conv}}(RNC_n)$ is a spectrahedron represented by a Hankel matrix $H_k(x) \succeq 0$ of size $k+1$ ( $k = \lfloor n/2 \rfloor$ ), attaining the optimal semidefinite extension degree (Scheiderer, 2024).

References

(Scheiderer, 2024) Convex hulls of curves in $n$ -space
(Scheiderer, 2012) Semidefinite representation for convex hulls of real algebraic curves
(0901.1821) Semidefinite representation of convex hulls of rational varieties
(Scheiderer, 2020) Second-order cone representation for convex subsets of the plane
(Wang et al., 2024) On semidefinite descriptions for convex hulls of quadratic programs
(Gouveia et al., 2010) Convex Hulls of Algebraic Sets
(Rostalski et al., 2010) Dualities in Convex Algebraic Geometry
(Helton et al., 2013) Matrix Convex Hulls of Free Semialgebraic Sets
(Harrison, 2015) Spectrahedra and Convex Hulls of Rank-One Elements
(Kalantari, 2019) A Triangle Algorithm for Semidefinite Version of Convex Hull Membership Problem
(2002.01566) On convex hulls of epigraphs of QCQPs
(Nie, 2011) Convex Hulls of Quadratically Parameterized Sets With Quadratic Constraints